ISAMA 2007, Texas A&M

1. ISAMA 2007, Texas A&M Carlo H. Séquin & Jaron Lanier CS Division & CET; College of Engineering University of California, Berkeley Hyper-Seeing the Regular Hendeca-choron . (= 11-Cell)

2. Jaron Lanier Visitor to the College of Engineering, U.C. Berkeleyand the Center for Entrepreneurship & Technology

3. “Do you know about the 4-dimensional 11-Cell ?-- a regular polytope in 4-D space;can you help me visualize that thing ?”Ref. to some difficult group-theoretic math paper Phone call from Jaron Lanier, Dec. 15, 2006

4. What Is a Regular Polytope ? • “Polytope”is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), “Polychoron” (4D), …to arbitrary dimensions. • “Regular”means: All the vertices, edges, faces, cells…are indistinguishable form each another. • Examples in 2D: Regular n-gons:

5. Regular Polyhedra in 3D The Platonic Solids: There are only 5. Why ? …

6. Why Only 5 Platonic Solids ? Lets try to build all possible ones: • from triangles: 3, 4, or 5 around a corner;  3 • from squares: only 3 around a corner;  1 . . . • from pentagons: only 3 around a corner;  1 • from hexagons:  planar tiling, does not close.  0 • higher N-gons:  do not fit around vertex without undulations (forming saddles)  now the edges are no longer all alike!

7. Let’s Build Some 4-D Polychora ... By analogy with 3-D polyhedra: • each will be bounded by 3-D cellsin the shape of some Platonic solid; • at every vertex (edge) the same numberof Platonic cells will join together; • that number has to be small enough,so that some wedge of free space is left, • which then gets forcibly closedand thereby produces some bending into 4-D.

8. All Regular Polychora in 4D Using Tetrahedra (70.5°): 3 around an edge (211.5°)  (5 cells) Simplex 4 around an edge (282.0°)  (16 cells) Cross polytope 5 around an edge (352.5°)  (600 cells) Using Cubes (90°): 3 around an edge (270.0°)  (8 cells) Hypercube Using Octahedra (109.5°): 3 around an edge (328.5°)  (24 cells) Hyper-octahedron Using Dodecahedra (116.5°): 3 around an edge (349.5°)  (120 cells) Using Icosahedra (138.2°):  NONE: angle too large (414.6°).

9. How to View a Higher-D Polytope ? For a 3-D object on a 2-D screen: • Shadow of a solid object is mostly a blob. • Better to use wire frame, so we can also see what is going on on the back side.

10. Oblique Projections • Cavalier Projection 3-D Cube  2-D 4-D Cube  3-D ( 2-D )

11. Projections: VERTEX/ EDGE /FACE/CELL- First. • 3-D Cube: Paralell proj. Persp. proj. • 4-D Cube: Parallel proj. Persp. proj.

12. Projections of a Hypercube to 3-D Cell-first Face-first Edge-first Vertex-first Use Cell-first: High symmetry; no coinciding vertices/edges

13. The 6 Regular Polytopes in 4-D

14. 120-Cell ( 600V, 1200E, 720F ) • Cell-first,extremeperspectiveprojection • Z-Corp. model

15. 600-Cell ( 120V, 720E, 1200F ) (parallel proj.) • David Richter

16. An 11-Cell ???Another Regular 4-D Polychoron ? • I have just shown that there are only 6. • “11” feels like a weird number;typical numbers are: 8, 16, 24, 120, 600. • The notion of a 4-D 11-Cell seems bizarre!

17. Kepler-Poinsot Solids 1 2 3 4 Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca • Mutually intersecting faces (all) • Faces in the form of pentagrams (3,4) But we can do even worse things ...

18. Q Hemicube (single-sided, not a solid any more!) 3 faces only vertex graph K4 3 saddle faces • If we are only concerned with topological connectivity, we can do weird things !

19. Hemi-dodecahedron connect oppositeperimeter points connectivity: Petersen graph six warped pentagons • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 9D space

20. Hemi-icosahedron connect oppositeperimeter points connectivity: graph K6 5-D Simplex;warped octahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 5D  THIS IS OUR BUILDING BLOCK !

21. Cross-cap Model of the Projective Plane • All these Hemi-polyhedra have the topology of the Projective Plane ...

22. Cross-cap Model of the Projective Plane • Has one self-intersection crease,a so called Whitney Umbrella

23. Another Model of the Projective Plane:Steiner’s Roman Surface • Has 6 Whitney umbrellas;tetrahedral symmetry. • Polyhedral model:An octahedronwith 4 tetrahedral faces removed, and 3 equatorial squares added.

24. Building Block: Hemi-icosahedron • The Projective Plane can also be modeled with Steiner’s Roman Surface. • This leads to a different set of triangles used(exhibiting more symmetry).

25. Gluing Two Steiner-Cells Together Hemi-icosahedron • Two cells share one triangle face • Together they use 9 vertices

26. + Cyan, Magenta= 5 Cells Adding More Cells . . . • Must never add more than 3 faces around an edge! 2 Cells + Yellow Cell= 3 Cells

27. 2 cells inner faces 3rd cell 4th cell 1 cell 5th cell Adding Cells Sequentially

28. How Much Further to Go ?? • So far we have assembled: 5 of 11 cells;but engaged all vertices and all edges,and 40 out of all 55 triangular faces! • It is going to look busy (messy)! • This object can only be “assembled”in your head ! You will not be able to “see” it !(like learning a city by walking around in it).

29. 4th white vertex used by next 3 cells (central) 11th vertex used by last 6 cells Two cells with cut-out faces A More Symmetrical Construction • Exploit the symmetry of the Steiner cell ! One Steiner cell 2nd cell added on “inside”

30. What is the Grand Plan ? • We know from:H.S.M. Coxeter: A Symmetrical Arrangement of Eleven Hemi-Icosahedra.Annals of Discrete Mathematics 20 (1984), pp 103-114. • The regular 4-D 11-Cellhas 11 vertices, 55 edges, 55 faces, 11 cells. • 3 cells join around every single edge. • Every pair of cells shares exactly one face.

31. The Basic Framework: 10-D Simplex • 10-D Simplex also has 11 vertices, 55 edges. • In 10-D space they can all have equal length. • 11-Cell uses only 55 of 165 triangular faces. • Make a suitable projection from 10-D to 3-D;(maintain as much symmetry as possible). • Select 11 different colors for the 11 cells;(Color faces with the 2 colors of the 2 cells).

32. The Complete Connectivity Diagram • From: Coxeter [2], colored by Tom Ruen

33. Symmetrical Arrangements of 11 Points 3-sided prism 4-sided prism 5-sided prism • Now just add all 55 edges and a suitable set of 55 faces.

34. 10 vertices on a sphere Point Placement Based on Plato Shells Same scheme as derived from the Steiner cell ! • Try for even more symmetry ! 1 + 4 + 6 vertices all 55 edges shown

35. The Full 11-Cell

36. Conclusions • The way to learn to “see” the hendecachoronis to try to understand its assembly process. • The way to do that is by pursuing several different approaches: • Bottom-up: understand the building-block cell,the hemi-icosahedron, and how a few of those fit together. • Top-down: understand the overall symmetry (K11),and the global connectivity of the cells. • An excellent application of hyper-seeing !

37. What Is the 11-Cell Good For ? • A neat mathematical object ! • A piece of “absolute truth”:(Does not change with style, new experiments) • A 10-dimensional building block …(Physicists believe Universe may be 10-D)

38. Are there More Polychora Like This ? • Yes – one more: the 57-Cell • Built from 57 Hemi-dodecahedra • 5 such single-sided cells join around edges • It is also self-dual: 57 V, 171 E, 171 F, 57 C. • I am still trying to get my mind around it . . .

39. Questions ? • Artistic coloring by Jaron Lanier

41. Building Block: Hemi-icosahedron • Uses all the edges of the 5D simplexbut only half of the available faces. • Has the topology of the Projective Plane(like the Cross-Cap ).